Thread: Zero content

A set z in R is said to have zero content if for any epsilon > 0, there is a finite collection of intervals I1, ...IL such that z is a subset of the union of those intervals (meaning all points in z are contained in the union of all the intervals) AND the sum of the lengths of the intervals is less than epsilon.

Thus, I was wondering if I could clarify something about zero content. So, for example, the set z = {y : y = x^2, x is part of [0, 10]} would not have zero content because for all the points z to be contained in intervals, the length of union of the intervals must be 10 and epsilon can be found which is less than 10.

However, am I right in thinking that the set z = {y : y = x^2, x is part of integers} has zero content? This is because arbitarily small intervals can be chosen around those points, and the union of the intervals can be made less than epsilon.

So basically, if f is continuous function, f(S) has zero content only when S is disconnected? Like sequences mapped by continuous functions have zero content because they are indexed by integers which form a disconnected set?

If is not constant then yes, because if has two elements say then, since is continous and is connected then is connected which means which means the content of .